The Monty Hall Problem

04 Feb The Monty Hall Problem

If you have been taking some maths tuition on statistics, you might have heard about the Monty Hall problem already. It is an extremely interesting problem. For those who are not familiar with statistics, fear not. It does not require any in-depth knowledge of statistics to understand the problem.

Introduction
The Monty Hall problem is a famous puzzle in statistics which has one of the most counter intuitive answers. Steve Selvin, a professor of biostatistics at UC Berkeley published a letter in the scientific journal “The American Statistician”. In this letter, he wrote about a problem loosely based on the game show “Let’s Make a Deal” and he dubbed the problem as “The Monty Hall” problem. Since then, the problem has achieved a cult status and has enjoyed immense popularity amongst maths enthusiasts.

The Problem
The Monty Hall problem goes like this. There are three closed doors marked as A,B and C. Behind each door, there is a prize. There are two goats behind two doors and behind one door, there is a car. You are now given an option to choose one of the three doors. Let’s say you pick door A. The game’s host Monty Hall, who knows what prize sits behind each door, will now open one of the remaining doors which has a goat. Let’s say he opens the door C which has a goat. Now Mr. Monty Hall comes up with a proposition. He offers you the option of changing your initial choice and swapping the doors. That is, if you wanted, you could choose door B and let go of door A. The Monty Hall problem asks what you should do in order to maximize your chance of winning the car. Should you take the option and swap doors or should you stick with your original choice? Before reading on, please take a moment to think about it.

The Analysis
It is most common mistake to think that swapping does not make any difference and your chances will remain the same. However, in order to maximize your chances of winning, you should swap every time you are given such a choice. In reality, if you swap, you are twice as likely to win – a whopping 100% increase in terms of probability. So how does this work? Let us see.

At first, you have 3 choices i.e. there are 3 doors that you can choose from. The car sits behind just one door. Those of you who have already taken some maths tuition on probability and statistics will be able to quickly point out the fact that the probability of you winning the car is 1/3 or around 33.33%. Similarly, the chance that you have chosen a door which has a goat behind it is 2/3 i.e. 66.66%. As you can see, when you choose for the first time, you are twice as likely to choose a door which has a goat behind it than choosing the one which has the car.

Next, Mr. Monty Hall reveals one of the doors which has a goat behind it. Since it was more likely that you had chosen a door with a goat (66.66&), if you switch, you will increase your chances of winning and in fact, you will be winning the car 66.66% of the time if you switch compared to the meagre 33.33% if you did not switch.

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